Editing Quadratic reciprocity (section)

===''q'' = ±1 and the first supplement===

Trivially 1 is a quadratic residue for all primes. The question becomes more interesting for −1. Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47. The former set of primes are all congruent to 1 modulo 4, and the latter are congruent to 3 modulo 4.

:'''First Supplement to Quadratic Reciprocity.''' The congruence <math>x^2 \equiv -1 \bmod{p}</math> is solvable if and only if <math>p</math> is congruent to 1 modulo 4.